Draft version July 5, 2021 Preprint typeset using LATEX style emulateapj v. 5/2/11 COMPOSITIONS AND ORIGINS OF OUTER PLANET SYSTEMS: INSIGHTS FROM THE ROCHE CRITICAL DENSITY Matthew S. Tiscareno1, Matthew M. Hedman1, Joseph A. Burns2;3, and Julie Castillo-Rogez4 1Center for Radiophysics and Space Research, Cornell University, Ithaca, NY 14853, USA. 2Department of Astronomy, Cornell University, Ithaca, NY 14853, USA. 3College of Engineering, Cornell University, Ithaca, NY 14853, USA. 4Jet Propulsion Laboratory, Pasadena, CA 91109, USA. Draft version July 5, 2021 ABSTRACT We consider the Roche critical density (ρRoche), the minimum density of an orbiting object that, at a given distance from its planet, is able to hold itself together by self-gravity. It is directly related to the more familiar \Roche limit," the distance from a planet at which a strengthless orbiting object of given density is pulled apart by tides. The presence of a substantial ring requires that transient clumps have an internal density less than ρRoche. Conversely, in the presence of abundant material for accretion, an orbiting object with density greater than ρRoche will grow. Comparing the ρRoche values at which the Saturn and Uranus systems transition rapidly from disruption-dominated (rings) to accretion-dominated (moons), we infer that the material composing Uranus' rings is likely more rocky, as well as less porous, than that composing Saturn's rings. From the high values of ρRoche at the innermost ring-moons of Jupiter and Neptune, we infer that those moons may be composed of denser material than expected, or more likely that they are interlopers that formed farther from their planets and have since migrated inward, now being held together by internal material strength. Finally, the \Portia group" of eight closely-packed Uranian moons has an overall surface density similar to that of Saturn's A ring. Thus, it can be seen as an accretion-dominated ring system, of similar character to the standard ring systems except that its material has a characteristic density greater than the local ρRoche. Subject headings: Planets and satellites: composition | Planets and satellites: dynamical evolution and stability | Planets and satellites: formation | Planets and satellites: rings 1. THE ROCHE LIMIT AND THE where R is radius and ρ is internal density, and the sub- ROCHE CRITICAL DENSITY script \p" denotes the central planet. The \Roche limit" is the distance from a planet within The dimensionless geometrical parameter γ = 4π=3 ≈ which its tides can pull apart a strengthless compact ob- 4:2 for a sphere, but is smaller for an object that takes ject. Simply speaking, a ring∗ would be expected to re- a non-spherical shape with its long axis pointing toward side inside its planet's Roche limit, while any disk of Saturn, as one would expect for an actively accreting material beyond that distance would be expected to ac- body and as observed for several of Saturn's ring-moons crete into one or more moons. However, the Roche limit (Porco et al. 2007; Charnoz et al. 2007). The region of does not actually have a single value, but depends par- gravitational dominance of a point mass moon, known as ticularly on the density, rotation, and internal material the \Roche lobe" or the \Hill sphere," is lemon-shaped, strength of the moon that may or may not get pulled with cusps at the two Lagrange points L1 and L2 (see, apart (Weidenschilling et al. 1984; Canup and Esposito e.g., Fig. 3.28 of Murray and Dermott 1999). Simply 1995). A simple value for the Roche limit of a spherical distributing the moon's material into the shape of its arXiv:1302.1253v1 [astro-ph.EP] 6 Feb 2013 moon, assuming no internal strength, can be calculated Roche lobe, with uniform density, yields γ ≈ 1:6 (Porco from a balance between the tidal force (i.e., the differ- et al. 2007). Going further for the case of an incom- ence between the planet's gravitational pull on one side pressible fluid, fully accounting for the feedback between of the moon and its pull on the other side) that would the moon's distorted shape and its (now non-point-mass) tend to pull a moon apart, and the moon's own grav- gravity field smooths out the cusps and further elongates ity that would tend to hold it together. This gives (e.g., the moon (Chandrasekhar 1969; Murray and Dermott Eq. 4.131 in Murray and Dermott 1999) 1999), leading to an even smaller value, γ ≈ 0:85. How- ever, some central mass concentration and the inability of a rubble pile with internal friction to exactly take its 4πρ 1=3 equilibrium shape will likely prevent γ from becoming a = R p ; (1) Roche p γρ quite this low. Hereafter, we will use the Porco et al. (2007) value of γ ≈ 1:6. Because the moon's internal density ρ appears in Eq. 1, ∗ Throughout this work, we use the word \ring" to refer to a substan- there is no single value of the Roche limit for a plane- tial annulus with sufficient material to support accretion. Our analysis tary system. This is intuitive, as a denser object can does not apply to tenuous structures such as Saturn's G and E rings. 2 a simple clump of material with no core cannot increase its internal density by shedding material, and thus must disjoin completely if its ρ is less than ρRoche by a sufficient margin.y Therefore, the fundamental criterion determin- ing whether a particular location in a planetary system is characterized by disruption (rings) or accretion (dis- crete moons) is whether the density naturally achieved by transient clumps is greater or less than ρRoche. 2. APPLICATIONS 2.1. A clue to composition As explained in the previous section, the persistent ex- istence of a ring at a given radial location a implies that the densities of transient clumps ρclump do not exceed ρRoche | that is, we expect that the composition of the ring material is such that ρclump . ρRoche. Because of Figure 1. The Roche critical density ρRoche (Eq. 2 or 3, with z γ = 1:6) plotted against planetary radii for Jupiter (red), Saturn the porosity inherent in any transient clump, we can ex- (cyan), Uranus (green), and Neptune (blue). An object must have pect ρclump to be only a fraction of ρsolid, the density of a density higher than ρRoche to be held together by its own gravity; a solid chunk of the material that composes the ring. conversely, in the presence of abundant disk material, an embedded object will actively accrete as long as its density remains higher As seen in Fig. 1, Saturn's main rings extend out- than ρRoche. The colored bars along the bottom show the extent ward to significantly lower values of ρRoche, approach- of each planet's main ring system. The bars along the left-hand ing 0.4 g cm−3, than are seen in any of the other three side are constructed from the bars along the bottom simply by the known ring systems, probably reflecting their much lower latter reflecting off the appropriately colored diagonal line. For each bar, a solid circle indicates the outermost extent, and the rock fraction (and higher fraction of water ice) as already corresponding minimum ρRoche, of the main rings. Figure from known from spectroscopy and photometry. In fact, Sat- Tiscareno (2013). urn's rings are composed almost entirely of water ice (Cuzzi et al. 2009), so we can conclude that ρclump is venture closer to the planet without danger of fragmen- ∼ 40% of ρsolid, at least in this case. Saturn's \ring tation than can an object that is less dense (see Fig. 1 moons" from Pan through Pandora, which range from of Weidenschilling et al. 1984). In fact, it is often more 4 to 40 km in mean radius and orbit in the vicinity of useful in the context of rings to consider the limit from the rings, have similar densities and porosities (Thomas planetary tides not as a critical distance but rather as a 2010). critical density. The compositions of the Uranian and Neptunian rings We can rearrange Eq. 1 such that, at any given distance are almost entirely unknown, as Voyager did not carry an infrared spectrometer with enough spatial resolution a from the planet, the Roche critical density ρRoche at which the moon's size entirely fills its Roche lobe is to detect them. However, it is clear from their low albedo that at least the surfaces of the ring particles cannot be 4πρp primarily water ice. Color imaging indicates that the ρRoche = 3 : (2) γ(a=Rp) Uranian rings are dark at all visible wavelengths, which would be consistent with the spectrum of carbon or or- 3 Then, substituting the planet's mass Mp = (4π=3)Rpρp, ganics, among other possibilities. we have Like at Saturn, the Uranus system has a clear 3M ρ = p : (3) boundary between disruption-dominated and accretion- Roche γa3 dominated regions (Fig. 2), notwithstanding some minor mixing at the boundary | at Uranus, Cordelia is in- While Eq. 3 gives the simplest method for calculating ward of the ring; at Saturn, Pan and Daphnis orbit ρRoche, Eq. 2 expresses its dependence on the distance in within the outermost parts of the A ring. The Uranian terms of planetary radii (a=Rp); when shown on a log-log x −3 main rings extend only to ρRoche ≈ 1:2 g cm . If we plot, as in Fig. 1, it appears as a straight line of slope assume a porosity for Uranian ring material with a sim- −3, normalized by each planet's internal density ρp.